Problem Analysis and Solutions:

The given function $f(x) = 2 \sin\left(\pi x + \frac{11\pi}{10}\right)$ is a transformation of $g(x) = \sin(x)$.

Part (a): Describe the transformations

1. Vertical Stretch: The coefficient $2$ before the sine function indicates a vertical stretch by a factor of 2. The amplitude of the sine wave is doubled.
2. Horizontal Compression: The coefficient $\pi$ inside the argument compresses the period of the sine wave. The period is affected and calculated later.
3. Phase Shift: The term $+\frac{11\pi}{10}$ inside the argument indicates a phase shift to the left by $\frac{11\pi}{10\pi} = \frac{11}{10}$ units.

Part (b): Find the period, amplitude, and phase shift

1. Amplitude: The amplitude is the absolute value of the vertical stretch factor, which is $|2| = 2$.
2. Period: The period is calculated using the formula: $\text{Period} = \frac{2\pi}{\text{coefficient of } x} = \frac{2\pi}{\pi} = 2$
3. Phase Shift: The phase shift is calculated as $-\frac{\text{constant added to } x}{\text{coefficient of } x}$: $\text{Phase Shift} = -\frac{\frac{11\pi}{10}}{\pi} = -\frac{11}{10}$ Thus, the phase shift is $\frac{11}{10}$ units to the left.

Part (c): Graphing one cycle of $f(x)$

To graph:

1. The period of the function is 2, so one cycle occurs from $x = -\frac{11}{10}$ (starting phase shift) to $x = -\frac{11}{10} + 2 = \frac{9}{10}$.
2. Divide the cycle into four equal intervals (quarter-periods) for key points:
   • Starting point: $x = -\frac{11}{10}$, $f(x) = 0$
   • First peak: $x = -\frac{11}{10} + \frac{1}{2} = -\frac{6}{10}$, $f(x) = 2$
   • Midpoint (zero): $x = -\frac{11}{10} + 1 = -\frac{1}{10}$, $f(x) = 0$
   • First trough: $x = -\frac{11}{10} + \frac{3}{2} = \frac{4}{10}$, $f(x) = -2$
   • End point: $x = -\frac{11}{10} + 2 = \frac{9}{10}$, $f(x) = 0$

I can plot this for you to visualize the transformation. Let me know if you'd like me to generate the graph for clarity. Would you like details about these steps or any related questions answered?

Questions to Explore Further:

1. How do transformations like vertical stretches affect the sine function's range?
2. What is the relationship between period and frequency in trigonometric functions?
3. How does a phase shift affect the symmetry of the graph?
4. What happens to the graph if the amplitude is a fraction (e.g., $\frac{1}{2}$)?
5. How would a negative amplitude (e.g., $-2$) affect the graph?

Tip:

When graphing trigonometric functions with transformations, always mark the amplitude, period, and phase shift clearly for accuracy